Information Percolation in Segmented Markets Darrell Duffie, - - PowerPoint PPT Presentation

Information Percolation in Segmented Markets

Darrell Duffie, Gustavo Manso, Semyon Malamud Stanford University, U.C. Berkeley, EPFL Probability, Control, and Finance In Honor of Ioannis Karatzas Columbia University, June, 2012

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Figure: An over-the-counter market.

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Cusip: 592646-AX-1

• 1%

0% 1% 2% 3% 4% 5% 1 2 3 4 5 6 7 8 9 10 Day Markup

Figure: Transaction price dispersion in muni market. Source: Green, Hollifield, and Sch¨ urhoff (2007). See, also, Goldstein and Hotchkiss (2007).

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Figure: Daily trade in the federal funds Market. Source: Bech and Atalay (2012).

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Information Transmission in Markets

Informational Role of Prices: Hayek (1945), Grossman (1976), Grossman and Stiglitz (1981).

◮ Centralized exchanges:

• Wilson (1977), Townsend (1978), Milgrom (1981), Vives (1993),

Pesendorfer and Swinkels (1997), and Reny and Perry (2006).

◮ Over-the-counter markets:

• Wolinsky (1990), Blouin and Serrano (2002), Golosov, Lorenzoni, and

Tsyvinski (2009).

• Duffie and Manso (2007), Duffie, Giroux, and Manso (2008), Duffie,

Malamud, and Manso (2010).

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Figure: Many OTC markets are dealer-intermediated.

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Model Primitives

◮ Agents: a non-atomic measure space (G, G, γ). ◮ Uncertainty: a probability space (Ω, F, P). ◮ An asset has a random payoff X with outcomes H and L. ◮ Agent i is initially endowed with a finite set Si = {s1, . . . , sn} of

{0, 1}-signals.

◮ Agents have disjoint sets of signals. ◮ The measurable subsets of Ω × G are enriched from the product

σ-algebra enough to allow signals to be essentially pairwise X-conditionally independent, and to allow Fubini, and thus the exact law of large numbers (ELLN). (Sun, JET, 2006).

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Information Types

After observing signals S = {s1, . . . , sn}, the logarithm of the likelihood ratio between states X = H and X = L is by Bayes’ rule: log P(X = H | s1, . . . , sn) P(X = L | s1, . . . , sn) = log P(X = H) P(X = L) +

n

• i=1

log pi(si | H) pi(si | L) , where pi(s | k) = P(si = s | X = k). We say that the “type” θ associated with this set of signals is θ =

n

• i=1

log pi(si | H) pi(si | L) .

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ELLN for Cross-Sectional Type Density

◮ The ELLN implies that, on the event {X = H}, the fraction of

agents whose initial type is no larger than some given number y is almost surely F H(y) =

• G

1{θα ≤ y} dγ(α) =

• G

P(θα ≤ y | X = H) dγ(α), where θα is the initial type of agent α.

◮ On the event {X = L}, the cross-sectional distribution function F L

• f types is likewise defined and characterized.

◮ We suppose that F H and F L have densities, denoted gH( · , 0) and

gL( · , 0) respectively.

◮ We write g(x, 0) for the random variable whose outcome is gH(x, 0)

• n the event {X = H} and gL(x, 0) on the event {X = L}.

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Information is Additive in Type

Proposition Let S = {s1, . . . , sn} and R = {r1, . . . , rm} be disjoint sets of signals, with associated types θ and φ. If two agents with types θ and φ reveal their types to each other, then both agents achieve the posterior type θ + φ. This follows from Bayes’ rule, by which log P(X = H | S, R, θ + φ) P(X = L | S, R, θ + φ) = log P(H = H) P(X = L) + θ + φ, = log P(X = H | θ + φ) P(X = L | θ + φ)

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Dynamics of Cross-Sectional Density of Types

Each period, each agent is matched, with probability λ, to a randomly chosen agent (uniformly distributed). They share their posteriors on X. Duffie and Sun (AAP 2007, JET 2012): With essential-pairwise-independent random matching of agents, g(x, t + 1) = (1 − λ)g(x, t) + +∞

−∞

λg(y, t)g(x − y, t) dy, x ∈ R, a.s. which can be written more compactly as g(t + 1) = (1 − λ)g(t) + λg(t) ∗ g(t), where ∗ denotes convolution.

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Solution of Cross-Sectional Distribution Types

◮ The Fourier transform of g( · , t) is

ˆ g(z, t) = 1 √ 2π +∞

−∞

e−izxg(x, t) dx.

◮ From (11), for each z in R,

d dt ˆ g(z, t) = −λˆ g(z, t) + λˆ g2(z, t), (1)

◮ Thus, the differential equation for the transform is solved by

ˆ g(z, t) = ˆ g(z, 0) eλt(1 − ˆ g(z, 0)) + ˆ g(z, 0). (2)

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Solution of Cross-Sectional Distribution Types

Proposition The unique solution of the dynamic equation (11) for the cross-sectional type density is the Wild sum g(θ, t) =

• n ≥ 1

e−λt(1 − e−λt)n−1g∗n(θ, 0), (3) where g∗n( · , 0) is the n-fold convolution of g( · , 0) with itself. The solution (3) is justified by noting that the Fourier transform ˆ g(z, t) can be expanded from (2) as ˆ g(z, t) =

• n≥1

e−λt(1 − e−λt)n−1ˆ g(z, 0)n, which is the transform of the proposed solution for g( · , t).

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Numerical Example

◮ Let λ = 1 and P(X = H) = 1/2. ◮ Agent α initially observes sα, with

P(sα = 1 | X = H) + P(sα = 1 | X = L) = 1.

◮ P(sα = 1 | X = H) has a cross-sectional distribution over investors

that is uniform over the interval [1/2, 1].

◮ On the event {X = H} of a high outcome, this initial allocation of

signals induces an initial cross-sectional density of f(p) = 2p for the likelihood P(X = H | sα) of a high state.

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On the event {X = H}, the evolution of the cross-sectional population density of posterior probabilities of the event {X = H}.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 t = 4.0 t = 3.0 t = 2.0 t = 1.0 t = 0.0

Conditional probability of high outcome. Population density

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Multi-Agent Meetings

The Boltzmann equation for the cross-sectional distribution µt of types is d dtµt = −λ µt + λ µ∗m

t

. We obtain the ODE, d dt ˆ µt = −λ ˆ µt + λ ˆ µm

t ,

whose solution satisfies ˆ µm−1

t

= ˆ µm−1 e(m−1)λt(1 − ˆ µm−1 ) + ˆ µm−1 . (4)

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Groups of 2 (blue) versus Groups of 3 (red)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y

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Groups of 2 (blue) versus Groups of 3 (red)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y

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Groups of 2 (blue) versus Groups of 3 (red)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y

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Groups of 2 (blue) versus Groups of 3 (red)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y

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Groups of 2 (blue) versus Groups of 3 (red)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y

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Groups of 2 (blue) versus Groups of 3 (red)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y

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Groups of 2 (blue) versus Groups of 3 (red)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y

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Other Extensions

◮ Privately gathered information. ◮ Public information releases (such as tweets or transaction

announcements).

• Duffie, Malamud, and Manso (2010).

◮ Endogenous search intensity

• Duffie, Malamud, and Manso (2009).

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A Segmented OTC Market

◮ Agents of class i ∈ {1, . . . , M} have matching probability λi. ◮ Upon meeting, the probability that a class-j agent is selected as a

counterparty is κij.

◮ At some time T, the economy ends, X is revealed, and the utility

realized by an agent of class i for each additional unit of the asset is Ui = vi1{X=L} + vH1{X=H}, for strictly positive vH and vi < vH.

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Trade by Seller’s Price Double Auction

◮ If vi = vj, there is no trade (Milgrom and Stokey, 1982;

Serrano-Padial, 2008).

◮ Upon a meeting with gains from trade, say vi < vj, the counterparties

participate in a seller’s price double auction.

◮ That is, if the buyer’s bid β exceeds the seller’s ask σ, trade occurs at

the price σ.

◮ The class of one’s counterparty is common knowledge.

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Equilibrium

The prices (σ, β) constitute an equilibrium for a seller of class i and a buyer of class j provided that, fixing β, the offer σ maximizes the seller’s conditional expected gain, E

• (σ − E(Ui | FS ∪ {β}))1{σ<β} | FS
• ,

and fixing σ, the bid β maximizes the buyer’s conditional expected gain E

• (E(Uj | FB ∪ {σ}) − σ)1{σ<β} | FB
• .

We look for equilibria that are completely revealing, of the form (B(θ), S(φ)), for a buyer and seller of types θ and φ, for strictly increasing B( · ) and S( · ).

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Technical Conditions

Definition: A function g( · ) on the real line is of exponential type α at −∞ if, for some constants c > 0 and γ > −1, lim

x → −∞

g(x) |x|γ eαx = c. (5) In this case, we write g(x) ∼ Exp−∞(c, γ, α). We use the notation g(x) ∼ Exp+∞(c, γ, α) analogously for the case of x → +∞. Condition: For all i, gi0 is C1 and strictly positive. For some α− ≥ 2.4 and α+ > 0 d dxgH

i0(x) ∼ Exp−∞(ci,−, γi,−, α−)

and d dxgH

i0(x) ∼ Exp+∞(ci,+, γi,+, α+)

for some ci,± > 0 and some γi,± ≥ 0.

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Equilibrium Bidding Strategies

◮ We provide an ODE for the equilibrium type Φ(b) of a prospective

buyer whose equilibrium bid is b. The ODE is the first-order condition for maximizing the probability of a trade multiplied by the expected profit given a trade.

◮ A prospective buyer of type φ bids B(φ) = Φ−1(φ). ◮ A prospective seller of type θ offers S(θ) = Θ−1(θ), where

Θ(v) = log v − vi vH − b − Φ(v), v ∈ (vi, vH) .

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The ODE for the Buyer’s Type

Lemma: For any initial condition φ0 ∈ R, there exists a unique solution Φ( · ) on [vi, vH) to the ODE Φ′(b) = 1 vi − vj b − vi vH − b 1 hH

it (Φ(b)) +

1 hL

it(Φ(b))

• ,

Φ(vi) = φ0. This solution, also denoted Φ(φ0, b), is monotone increasing in both b and φ0. Further, limb→vH Φ(b) = +∞ . The limit Φ(−∞, b) = limφ0→−∞ Φ(φ0, b) exists and is strictly monotone and continuously differentiable with respect to b.

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Bidding Strategies

Proposition Suppose that (S, B) is a continuous equilibrium such that S(θ) ≤ vH for all θ ∈ R. Let φ0 = B−1(vi) ≥ −∞. Then, B(φ) = Φ−1(φ), φ > φ0, Further, limθ→−∞ S(θ) = vi and limθ→−∞ S(θ) = vH, and for any θ, we have S(θ) = Θ−1(θ). Any buyer of type φ < φ0 does not trade, and has a bidding policy B that is not uniquely determined at types below φ0. The unique welfare maximizing equilibrium is that associated with limφ0→−∞ Φ(φ0, b). This equilibrium exists and is fully revealing.

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Evolution of Type Distribution

Dynamics for the distribution of types of agents of class i: gi,t+1 = (1 − λi) git + λi git ∗

M

• j=1

κij gjt, i ∈ {1, . . . , M} . Taking Fourier transforms: ˆ gi,t+1 = (1 − λi) ˆ git + λi ˆ git

M

• j=1

κij ˆ gjt, i ∈ {1, . . . , M}.

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Evolution of Type Distribution

Dynamics for the distribution of types of agents of class i: gi,t+1 = (1 − λi) git + λi git ∗

M

• j=1

κij gjt, i ∈ {1, . . . , M} . Taking Fourier transforms: ˆ gi,t+1 = (1 − λi) ˆ git + λi ˆ git

M

• j=1

κij ˆ gjt, i ∈ {1, . . . , M}.

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Special Case: N = 2 and λ1 = λ2

Proposition: Suppose N = 2 and λ1 = λ2 = λ. Then ˆ ψ1 = e−λt ( ˆ ψ20 − ˆ ψ10) ˆ ψ20e− ˆ

ψ20(1−e−λt) − ˆ

ψ10e− ˆ

ψ10(1−e−λt)

ˆ ψ10 e− ˆ

ψ10(1−e−λt)

ˆ ψ2 = e−λt ( ˆ ψ20 − ˆ ψ10) ˆ ψ20e− ˆ

ψ20(1−e−λt) − ˆ

ψ10e− ˆ

ψ10(1−e−λt)

ˆ ψ20 e− ˆ

ψ20(1−e−λt).

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General Case: Wild Sum Representation

Theorem: There is a unique solution of the evolution equation, given by ψit =

• k∈ZN

+

ait(k) ψ∗k1

10 ∗ · · · ∗ ψ∗kN N0 ,

where ψ∗n

i0 denotes n-fold convolution,

a′

it = −λi ait + λi ait ∗ N

• j=1

κij ajt, ai0 = δei, (ait ∗ ajt)(k1, . . . , kN) =

• l=(l1,...,lN) ∈ ZN

+ , l<k

ait(l) ajt(k − l), and ait(ei) = e−λit ai0(ei).

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Endogenous Information Acquisition

◮ A signal packet is a set of signals with type density f, satisfying the

technical conditions.

◮ Agents are endowed with Nmin signal packets, and can acquire up to

n more, at a cost of π each.

◮ Agents conjecture the packet quantity choices N = (N1, . . . , NM) of

the M classes.

◮ An agent of class i who initially acquires n signal packets, and as a

result has information-type Θn,N,t at time t, has initial expected utility ui,n,N = E  −πn +

T

• t=1

λi

• j

κij vijt(Θn,N,t; Bijt, Sijt)   . (6)

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Information Acquisition Equilibrium

Some (Ni, (Sijt, Bijt), git) is a pure-strategy rational expectations equilibrium if

◮ The cross-sectional type density git satisfies the evolution equation

with initial condition the (Nmin + Ni)-fold convolution of the packet type density f.

◮ The bid and ask functions (Sijt, Bijt) are revealing double-auction

equilibria.

◮ The number Ni of signal packets acquired by class i solves

max n ∈ {0,...,n} ui,n,N.

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3-Class Incentives

◮ We take the case of two equal-mass seller classes with matching

probabilities λ1 and λ2 > λ1.

◮ The buyer class has matching probability (λ1 + λ2)/2. ◮ d dxfH(x) ∼ Exp−∞(c0, 0, α + 1) and d dxfH(x) ∼ Exp+∞(c0, 0, −α) for some α ≥ 1.4 for some c0 > 0.

Proposition For

vb−vs vH−vb and T large enough, information acquisition is a strategic

• complement. By contrast, for smaller T, there exist counterexamples to

strategic complementarity.

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Information Acquisition Incentives

◮ If T is not too great, increasing λ2 of the more active class-2 sellers

lowers the incentive of the less active class-1 sellers to gather more

• information. This can be explained as follows.

◮ As class-2 sellers become more active, buyers learn at a faster rate.

The impact of this on the incentive of the “slower” class-1 sellers to gather information is determined by a “learning effect” and an

• pposing “pricing effect.”

◮ The learning effect is that, knowing that buyers will learn faster as λ2

is raised, a less connected seller is prone to acquire more information in order to avoid being at an informational disadvantage when facing buyers.

◮ The pricing effect is that, in order to avoid missing unconditional

private-value expected gains from trade with better-informed buyers, sellers find it optimal to reduce their ask prices.

◮ The learning effect dominates the pricing effect if and only if there are

sufficiently many trading rounds.

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◮ We show cases in which increasing λ2 leads to a full collapse of

information acquisition (meaning that, in any equilibrium, the fraction

• f agents that acquire signals is zero).

◮ Compare with the case of a static double auction, corresponding to

T = 0. With only one round of trade, the learning effect is absent and the expected gain from acquiring information for class-1 sellers is proportional to λ1 and does not depend on λ2. Similarly, the gain from information acquisition for buyers is linear and increasing in λ2. Consequently, in the static case, an increase in λ2 always leads to more information acquisition.

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